Multivariate Regression

Multivariate Regression

Real datasets have multiple features. Multivariate linear regression extends the single-variable case to $d$ input features:

$\hat{y} = \mathbf{w} \cdot \mathbf{x} + b = \sum_{j=1}^{d} w_j x_j + b$

where $\mathbf{w} = [w_1, \ldots, w_d]$ is the weight vector and $\mathbf{x} = [x_1, \ldots, x_d]$ is the feature vector.

Dot Product

The core operation is the dot product:

$\mathbf{w} \cdot \mathbf{x} = \sum_{j=1}^{d} w_j x_j$

Multivariate MSE

For a dataset $\{(\mathbf{x}^{(i)}, y^{(i)})\}_{i=1}^n$:

$\text{MSE} = \frac{1}{n} \sum_{i=1}^{n} \left( \hat{y}^{(i)} - y^{(i)} \right)^2$

where each $\hat{y}^{(i)} = \mathbf{w} \cdot \mathbf{x}^{(i)} + b$.

Your Task

Implement:

• dot_product(x, w) — sum of element-wise products
• predict_multi(x, w, b) — dot product plus bias
• mse_multi(X, w, b, y_true) — MSE over a dataset of feature vectors